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We conjecture that the free-fermion part of the eigenspectrum observedrecently for the $SU_q(N)$ Perk-Schultz spin chain Hamiltonian in a finitelattice with $q=\exp (i\pi (N-1)/N)$ is a consequence of the existence of aspecial simple eigenvalue for the transfer matrix of the auxiliaryinhomogeneous $SU_q(N-1)$ vertex model which appears in the nested Bethe ansatzapproach. We prove that this conjecture is valid for the case of the SU(3) spinchain with periodic boundary condition. In this case we obtain a formula forthe components of the eigenvector of the auxiliary inhomogeneous 6-vertex model($q=\exp (2 i \pi/3)$), which permit us to find one by one all components ofthis eigenvector and consequently to find the eigenvectors of the free-fermionpart of the eigenspectrum of the SU(3) spin chain. Similarly as in the knowncase of the $SU_q(2)$ case at $q=\exp(i2\pi/3)$ our numerical and analyticalstudies induce some conjectures for special rates of correlation functions.Comment: 25 pages and no figure

The wavefunctions for the free-fermion part of the spectrum of theSUq(N) quantum spin models

The wavefunctions for the free-fermion part of the spectrum of theSU_q(N) quantum spin models